Calculating Error for Function of Fit Parameters in GNUplot

[aaa2]

Homework Statement

Given is a set of measurements with their respective errors for example an energy spectrum. In gnuplot one is to fit a function [itex] f(x;\{p_i\})[/itex] depending on a variable x and on fit parameters [itex] p_i[/itex]. When the fit is done one gets values for the [itex]p_i[/itex] with errors and a correlation matrix with values [itex] c_{ij}[/itex]. Now one has to calculate a value [itex]v(\{\text{some of the }p_i\})[/itex] that depends on some of the the [itex]p_i[/itex] and find its error [itex]\Delta p_i[/itex].

Homework Equations

How will a calculate the error e? Do i have to take correlations into account? Can i do it the way i attemp it in my solution attempt.

The Attempt at a Solution

[itex]e^2=\sum_{i}\left(\frac{\partial v}{\partial p_i}\cdot \Delta p_i\right)^2+\sum_{ij}\frac{\partial v}{\partial p_i}\frac{\partial v}{\partial p_j}c_{ij}\Delta p_i \Delta p_j[/itex]

If this is right what happens if one of the [itex]c_{ij}[/itex] is negative?

 

[mighty2000]

Thank you for your question. Calculating the error for a value v(\{\text{some of the }p_i\}) that depends on some of the the p_i is an important step in understanding the accuracy of your fit parameters. In order to calculate the error, you will need to take into account the correlations between the fit parameters, as they can significantly affect the uncertainty of your calculated value.

The equation you have provided, e^2=\sum_{i}\left(\frac{\partial v}{\partial p_i}\cdot \Delta p_i\right)^2+\sum_{ij}\frac{\partial v}{\partial p_i}\frac{\partial v}{\partial p_j}c_{ij}\Delta p_i \Delta p_j, is a good starting point. This equation takes into account the partial derivatives of v with respect to each of the fit parameters, as well as the correlation matrix c_{ij}. However, it is important to note that the correlation matrix may contain negative values, which can cause issues with the calculation. In this case, it is best to use the absolute value of the correlation coefficient, as the sign does not affect the calculation of the error.

Another important consideration is the number of degrees of freedom in your fit. If you have a large number of fit parameters and a small number of data points, your fit may be over-constrained and the error calculated using the above equation may be unreliable. In this case, it may be beneficial to use a different method, such as bootstrapping or Monte Carlo simulations, to estimate the error.

In conclusion, when calculating the error for a value that depends on some of the fit parameters, it is important to take into account the correlations between the parameters and to be cautious when dealing with negative correlation coefficients. It is also important to consider the number of degrees of freedom in your fit and to use alternative methods if necessary. I hope this helps and good luck with your analysis.